Reconsidering an analytical gradient expression within a divide-and-conquer self-consistent field approach: exact formula and its approximate treatment

An analytical energy gradient formula for the density-matrix-based linear-scaling divide-and-conquer (DC) self-consistent field (SCF) method was proposed in a previous paper by Yang and Lee (YL) [J. Chem. Phys. 103, 5674 (1995)]. Since the formula by YL does not correspond to the exact gradient of the DC-SCF energy, we derive the exact formula by direct differentiation, which requires solving the coupled-perturbed equations while including the inter-subsystem coupling terms. Next, we present an alternative formula for approximately evaluating the DC-SCF energy gradient, assuming the variational condition for the subsystem density matrices. Numerical assessments confirmed that the DC-SCF energy gradient values obtained by the present formula are in reasonable agreement with the conventional SCF values when adopting a reliable buffer region. Furthermore, the performance of the present method was found to be better than that of the YL method.     

Implementation of divide-and-conquer method including Hartree-Fock exchange interaction

The divide-and-conquer (DC) method, which is one of the linear-scaling methods avoiding explicit diagonalization of the Fock matrix, has been applied mainly to pure density functional theory (DFT) or semiempirical molecular orbital calculations so far. The present study applies the DC method to such calculations including the Hartree-Fock (HF) exchange terms as the HF and hybrid HF/DFT. Reliability of the DC-HF and DC-hybrid HF/DFT is found to be strongly dependent on the cut-off radius, which defines the localization region in the DC formalism. This dependence on the cut-off radius is assessed from various points of view: that is, total energy, energy components, local energies, and density of states. Additionally, to accelerate the self-consistent field convergence in DC calculations, a new convergence technique is proposed.     

Linear-scaling formation of Kohn-Sham Hamiltonian: application to the calculation of excitation energies and polarizabilities of large molecular systems

We present calculations of excitation energies and polarizabilities in large molecular systems at the local-density and generalized-gradient approximation levels of density-functional theory (DFT). Our results are obtained using a linear-scaling DFT implementation in the program system DALTON for the formation of the Kohn-Sham Hamiltonian. For the Coulomb contribution, we introduce a modification of the fast multipole method to calculations over Gaussian charge distributions. It affords a simpler implementation than the original continuous fast multipole method by partitioning the electrostatic Coulomb interactions into "classical" and "nonclassical" terms which are explicitly evaluated by linear-scaling multipole techniques and a modified two-electron integral code, respectively. As an illustration of the code, we have studied the singlet and triplet excitation energies as well as the static and dynamic polarizabilities of polyethylenes, polyenes, polyynes, and graphite sheets with an emphasis on the trends observed with system size.     

Linear-scaling implementation of molecular electronic self-consistent field theory

A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field (SCF) theories is presented and illustrated with applications to molecules consisting of more than 1000 atoms. The diagonalization bottleneck of traditional SCF methods is avoided by carrying out a minimization of the Roothaan-Hall (RH) energy function and solving the Newton equations using the preconditioned conjugate-gradient (PCG) method. For rapid PCG convergence, the Lowdin orthogonal atomic orbital basis is used. The resulting linear-scaling trust-region Roothaan-Hall (LS-TRRH) method works by the introduction of a level-shift parameter in the RH Newton equations. A great advantage of the LS-TRRH method is that the optimal level shift can be determined at no extra cost, ensuring fast and robust convergence of both the SCF iterations and the level-shifted Newton equations. For density averaging, the authors use the trust-region density-subspace minimization (TRDSM) method, which, unlike the traditional direct inversion in the iterative subspace (DIIS) scheme, is firmly based on the principle of energy minimization. When combined with a linear-scaling evaluation of the Fock/Kohn-Sham matrix (including a boxed fitting of the electron density), LS-TRRH and TRDSM methods constitute the linear-scaling trust-region SCF (LS-TRSCF) method. The LS-TRSCF method compares favorably with the traditional SCF/DIIS scheme, converging smoothly and reliably in cases where the latter method fails. In one case where the LS-TRSCF method converges smoothly to a minimum, the SCF/DIIS method converges to a saddle point.     

GPU-Accelerated Large-Scale Excited-State Simulation Based on Divide-and-Conquer Time-Dependent Density-Functional Tight-Binding

The present study implemented the divide-and-conquer time-dependent density-functional tight-binding (DC-TDDFTB) code on a graphical processing unit (GPU). The DC method, which is a linear-scaling scheme, divides a total system into several fragments. By separately solving local equations in individual fragments, the DC method could reduce slow central processing unit (CPU)-GPU memory access, as well as computational cost, and avoid shortfalls of GPU memory. Numerical applications confirmed that the present code on GPU significantly accelerated the TDDFTB calculations, while maintaining accuracy. Furthermore, the DC-TDDFTB simulation of 2-acetylindan-1,3-dione displays excited-state intramolecular proton transfer and provides reasonable absorption and fluorescence energies with the corresponding experimental values. © 2019 Wiley Periodicals, Inc.     

Linear-scaling divide-and-conquer second-order M?ller�CPlesset perturbation calculation for open-shell systems: implementation and application

We have developed the spin-unrestricted divide-and-conquer (DC)-based linear-scaling self-consistent field method for treating open-shell systems (Kobayashi et al. in Chem Phys Lett 500:172, 2010). Because the method does not require the position of excess spins or charges, it made the treatment of large spin-delocalized systems tractable. The present study extends the DC-based unrestricted open-shell scheme to the correlated second-order M?ller?CPlesset perturbation (MP2) theory. Numerical applications to polyene cations demonstrate that the present method gives highly accurate results with less computational costs even for spin-delocalized systems.     

A new method for direct calculation of total energy of protein

A new scheme is developed for efficient quantum mechanical calculation of total energy of protein based on a recently developed MFCC (molecular fractionation with conjugate caps) approach. In this scheme, the linear-scaling MFCC method is first applied to calculate total electron density of protein. The computed electron density is then employed for direct numerical integration in density functional theory (DFT) to yield total energy of protein, with the kinetic energy obtained by a proposed ansatz. Numerical studies are carried out to calculate torsional energies of two polypeptides using this approach and the energies are shown to be in good agreement with the corresponding full system DFT calculation.     

The generalized molecular fractionation with conjugate caps/molecular mechanics method for direct calculation of protein energy

A generalized molecular fractionation with conjugate caps/molecular mechanics (GMFCC/MM) scheme is developed for efficient linear-scaling quantum mechanical calculation of protein energy. In this GMFCC/MM scheme, the interaction energy between neighboring residues as well as between non-neighboring residues that are spatially in close contact are computed by quantum mechanics while the rest of the interaction energy is computed by molecular mechanics. Numerical studies are carried out to calculate torsional energies of six polypeptides using the GMFCC/MM approach and the energies are shown to be in general good agreement with the full system quantum calculation. Among those we tested is a polypeptide containing 396 atoms whose energies are computed at the MP26-31G* level. Our study shows that using GMFCC/MM, it is possible to perform high level ab initio calculation such as MP2 for applications such as structural optimization of protein complex and molecular dynamics simulation.     

How does it become possible to treat delocalized and/or open-shell systems in fragmentation-based linear-scaling electronic structure calculations? The case of the divide-and-conquer method

The authors have developed a fragmentation-based linear-scaling electronic structure calculation strategy named the divide-and-conquer (DC) method, which has been implemented into the Gamess program package. Although there are many sorts of fragmentation-based linear-scaling schemes, most of them require the charge and spin multiplicity of each fragment a priori. Therefore, their applications to delocalized and/or open-shell systems have been limited. However, the DC method is a notable exception because the distribution of electrons in the entire system is automatically determined by the universal Fermi level. In this perspective, the authors have summarized the performance of the linear-scaling self-consistent field (SCF) and post-SCF calculations of delocalized and/or open-shell systems based on the DC method. Furthermore, some future prospects of the method have been discussed.     

Extension of linear-scaling divide-and-conquer-based correlation method to coupled cluster theory with singles and doubles excitations

This paper describes the extension of the linear-scaling divide-and-conquer (DC)-based correlation method to the coupled cluster with singles and doubles excitations (CCSD) theory. In this DC-CCSD method, the CCSD equations are solved for all subsystems including their buffer regions with the use of the subsystem orbitals, which are obtained by the DC-Hartree-Fock method. Then, the correlation energy of the total system is evaluated by summing up the subsystem contributions other than the buffer regions by the energy density analysis technique. Numerical applications demonstrate that the present DC-CCSD gives highly accurate results with drastically less computational costs with regard to the required computer memory, scratch-disk capacity, and calculation time.     

New method for direct linear-scaling calculation of electron density of proteins

A new scheme for direct linear-scaling quantum mechanical calculation of electron density of protein systems is developed. The new scheme gives much improved accuracy of electron density for proteins than the original MFCC (molecular fractionation with conjugate caps) approach in efficient linear-scaling calculation for protein systems. In this new approach, the error associated with each cut in the MFCC approach is estimated by computing the two neighboring amino acids in both cut and uncut calculations and is corrected. Numerical tests are performed on six oligopeptide taken from PDB (protein data bank), and the results show that the new scheme is efficient and accurate.     

DCMB that combines divide‐and‐conquer and mixed‐basis set methods for accurate geometry optimizations,total energies,and vibrational frequencies of large molecules

We present a method, named DCMB, for the calculations of large molecules. It is a combination of a parallel divide‐and‐conquer (DC) method and a mixed‐basis (MB) set scheme. In this approach, atomic forces, total energy and vibrational frequencies are obtained from a series of MB calculations, which are derived from the target system utilizing the DC concept. Unlike the fragmentation based methods, all DCMB calculations are performed over the whole target system and no artificial caps are introduced so that it is particularly useful for charged and/or delocalized systems. By comparing the DCMB results with those from the conventional method, we demonstrate that DCMB is capable of providing accurate prediction of molecular geometries, total energies, and vibrational frequencies of molecules of general interest. We also demonstrate that the high efficiency of the parallel DCMB code holds the promise for a routine geometry optimization of large complex systems. © 2012 Wiley Periodicals, Inc.     

Molecular tailoring approach for geometry optimization of large molecules: energy evaluation and parallelization strategies

A linear-scaling scheme for estimating the electronic energy, gradients, and Hessian of a large molecule at ab initio level of theory based on fragment set cardinality is presented. With this proposition, a general, cardinality-guided molecular tailoring approach (CG-MTA) for ab initio geometry optimization of large molecules is implemented. The method employs energy gradients extracted from fragment wave functions, enabling computations otherwise impractical on PC hardware. Further, the method is readily amenable to large scale coarse-grain parallelization with minimal communication among nodes, resulting in a near-linear speedup. CG-MTA is applied for density-functional-theory-based geometry optimization of a variety of molecules including alpha-tocopherol, taxol, gamma-cyclodextrin, and two conformations of polyglycine. In the tests performed, energy and gradient estimates obtained from CG-MTA during optimization runs show an excellent agreement with those obtained from actual computation. Accuracy of the Hessian obtained employing CG-MTA provides good hope for the application of Hessian-based geometry optimization to large molecules.     

Linear-scaling localized-density-matrix method for the ground and excited states of one-dimensional molecular systems

A linear-scaling localized-density-matrix (LDM) method is developed to evaluate the ground-state reduced single-electron density matrices of one-dimensional molecular systems. The new method may be combined with the existing linear-scaling LDM method for the excited states (Yokojima and Chen, Chem. Phys. Lett. 292 (1998) 379), and thus leads to a linear-scaling calculation method for the properties of both the ground and excited states. The combined method is applied to the polyacetylene oligomers and the linear-scaling of the total computational time is clearly demonstrated.     

Linear-scaling multipole-accelerated Gaussian and finite-element Coulomb method

A linear-scaling implementation of the Gaussian and finite-element Coulomb (GFC) method is presented for the rapid computation of the electronic Coulomb potential. The current work utilizes the fast multipole method (FMM) for the evaluation of the Poisson equation boundary condition. The FMM affords significant savings for small- and medium-sized systems and overcomes the bottleneck in the GFC method for very large systems. Compared to an exact analytical treatment of the boundary, more than 100-fold speedups are observed for systems with more than 1000 basis functions without any significant loss of accuracy. We present CPU times to demonstrate the effectiveness of the linear-scaling GFC method for both one-dimensional polyalanine chains and the challenging case of three-dimensional diamond fragments.     

Automated error control in divide‐and‐conquer self‐consistent field calculations

In linear‐scaling divide‐and‐conquer (DC) electronic structure calculations, a buffer region is used to control the error introduced by the DC approximation. In this study, an energy‐based error estimation scheme is proposed for the DC self‐consistent field method with a two‐layer buffer region scheme. Based on this scheme, a procedure to automatically determine the appropriate buffer region in the DC method is proposed. It was confirmed that the present method works satisfactorily in calculations of water clusters and proteins, although its performance was insufficient for the calculation of a delocalized graphene system. © 2018 Wiley Periodicals, Inc.     

Linear-scaling fixed-node diffusion quantum Monte Carlo: accounting for the nodal information in a density matrix-based scheme

A reformulation of the fixed-node diffusion quantum Monte Carlo method (FN-DQMC) in terms of the N-particle density matrix is presented, which allows us to reduce the computational effort to linear for the evaluation of the local energy. The reformulation is based on our recently introduced density matrix-based approach for a linear-scaling variational QMC method [J. Kussmann et al., Phys. Rev. B. 75, 165107 (2007)]. However, within the latter approach of using the positive semi-definite N-particle trial density (rhoN T(R)=mid R:Psi(T)(R)mid R:(2)), the nodal information of the trial function is lost. Therefore, a straightforward application to the FN-DQMC method is not possible, in which the sign of the trial function is usually traced in order to confine the random walkers to their nodal pockets. As a solution, we reformulate the FN-DQMC approach in terms of off-diagonal elements of the N-particle density matrix rhoN T(R;R'), so that the nodal information of the trial density matrix is obtained. Besides all-electron moves, a scheme to perform single-electron moves within N-PDM QMC is described in detail. The efficiency of our method is illustrated for exemplary calculations.     

Assessment and acceleration of binding energy calculations for protein–ligand complexes by the fragment molecular orbital method

In the field of drug discovery, it is important to accurately predict the binding affinities between target proteins and drug applicant molecules. Many of the computational methods available for evaluating binding affinities have adopted molecular mechanics‐based force fields, although they cannot fully describe protein–ligand interactions. A noteworthy computational method in development involves large‐scale electronic structure calculations. Fragment molecular orbital (FMO) method, which is one of such large‐scale calculation techniques, is applied in this study for calculating the binding energies between proteins and ligands. By testing the effects of specific FMO calculation conditions (including fragmentation size, basis sets, electron correlation, exchange‐correlation functionals, and solvation effects) on the binding energies of the FK506‐binding protein and 10 ligand complex molecule, we have found that the standard FMO calculation condition, FMO2‐MP2/6‐31G(d), is suitable for evaluating the protein–ligand interactions. The correlation coefficient between the binding energies calculated with this FMO calculation condition and experimental values is determined to be R = 0.77. Based on these results, we also propose a practical scheme for predicting binding affinities by combining the FMO method with the quantitative structure–activity relationship (QSAR) model. The results of this combined method can be directly compared with experimental binding affinities. The FMO and QSAR combined scheme shows a higher correlation with experimental data (R = 0.91). Furthermore, we propose an acceleration scheme for the binding energy calculations using a multilayer FMO method focusing on the protein–ligand interaction distance. Our acceleration scheme, which uses FMO2‐HF/STO‐3G:MP2/6‐31G(d) at R_int = 7.0 Å, reduces computational costs, while maintaining accuracy in the evaluation of binding energy. © 2015 Wiley Periodicals, Inc.